Editing Moore–Penrose inverse (section)

==Definition==
For <math>A \in \mathbb{K}^{m\times n}</math>, a pseudoinverse of {{mvar| A}} is defined as a matrix {{tmath| A^+ \in \mathbb{K}^{n\times m} }} satisfying all of the following four criteria, known as the Moore–Penrose conditions:<ref name="Penrose1955"/><ref name="GvL1996">{{cite book | last=Golub | first=Gene H. | author-link=Gene H. Golub |author2=Charles F. Van Loan | title=Matrix computations | url=https://archive.org/details/matrixcomputatio00golu_910 | url-access=limited | edition=3rd | publisher=Johns Hopkins | location=Baltimore | year=1996 | isbn=978-0-8018-5414-9 | pages = [https://archive.org/details/matrixcomputatio00golu_910/page/n283 257]–258| author2-link=Charles F. Van Loan }}</ref>
# {{tmath| A A^+ }} need not be the general identity matrix, but it maps all column vectors of {{mvar| A }} to themselves: <math display="block">A A^+ A = \; A.</math>
# {{tmath| A^+ }} acts like a [[weak inverse]]: <math display="block">A^+ A A^+ = \; A^+.</math>
# {{tmath| A A^+ }} is [[Hermitian matrix|Hermitian]]: <math display="block">\left(A A^+\right)^* = \; A A^+.</math>
# {{tmath| A^+A }} is also Hermitian: <math display="block">\left(A^+ A\right)^* = \; A^+ A.</math>

Note that <math>A^+A</math> and <math>AA^+</math> are idempotent operators, as follows from <math>(AA^+)^2=A A^+</math> and <math>(A^+ A)^2=A^+ A</math>. More specifically, <math>A^+A</math> projects onto the image of <math>A^T</math> (equivalently, the span of the rows of <math>A</math>), and <math>AA^+</math> projects onto the image of <math>A</math> (equivalently, the span of the columns of <math>A</math>). In fact, the above four conditions are fully equivalent to <math>A^+A</math> and <math>AA^+</math> being such orthogonal projections: <math>AA^+</math> projecting onto the image of <math>A</math> implies <math>(A A^+)A=A</math>, and <math>A^+A</math> projecting onto the image of <math>A^T</math> implies <math>(A^+A)A^T=A^T</math>.

The pseudoinverse <math>A^+</math> exists for any matrix <math>A \in \mathbb{K}^{m\times n}</math>. If furthermore <math>A</math> is full [[rank (linear algebra)|rank]], that is, its rank is {{tmath| \min \{ m,n \} }}, then {{tmath| A^+ }} can be given a particularly simple algebraic expression. In particular:

* When {{tmath| A }} has linearly independent columns (equivalently, {{tmath| A }} is injective, and thus {{tmath| A^* A }} is invertible), {{tmath| A^+ }} can be computed as<math display="block">A^+ = \left(A^* A\right)^{-1} A^*.</math>This particular pseudoinverse is a ''left inverse'', that is, <math>A^+A = I</math>.
* If, on the other hand, <math>A</math> has linearly independent rows (equivalently, <math>A</math> is surjective, and thus {{tmath| A A^* }} is invertible), {{tmath| A^+ }} can be computed as<math display="block">A^+ = A^* \left(A A^*\right)^{-1}.</math>This is a ''right inverse'', as <math>A A^+ = I</math>.

In the more general case, the pseudoinverse can be expressed leveraging the [[singular value decomposition]]. Any matrix can be decomposed as <math> A=UDV^*</math> for some isometries <math>U,V</math> and diagonal nonnegative real matrix <math>D</math>. The pseudoinverse can then be written as <math>A^+=V D^{+} U^*</math>, where <math>D^{+}</math> is the pseudoinverse of <math>D</math> and can be obtained by transposing the matrix and replacing the nonzero values with their multiplicative inverses.{{sfn|Campbell|Meyer|1991}} That this matrix satisfies the above requirement is directly verified observing that <math>AA^+=UU^*</math> and <math>A^+ A=VV^*</math>, which are the projections onto image and support of <math>A</math>, respectively.